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 A008919 Numbers k such that k written backwards is a nontrivial multiple of k. 13
 1089, 2178, 10989, 21978, 109989, 219978, 1099989, 2199978, 10891089, 10999989, 21782178, 21999978, 108901089, 109999989, 217802178, 219999978, 1089001089, 1098910989, 1099999989, 2178002178, 2197821978, 2199999978, 10890001089 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS There are 2*Fibonacci(floor((n-2)/2)) terms with n digits (this is A214927 or essentially twice A103609). - N. J. A. Sloane, Mar 20 2013 All terms are made of "symmetric" concatenations of 1089 and/or 2178, with an arbitrary numbers of 9's inserted in the middle of these and 0's inserted between them. See A031877 for the reversals and further information: union of the two, sequences "made of" 1089 or 2178 only. - M. F. Hasler, Jun 23 2019 Also: 99 times A061852: numbers that are palindromic, have only digits in {0, 1} or in {0, 2}, and no isolated ("single") digit. - M. F. Hasler, Oct 17 2022 REFERENCES W. W. R. Ball and H. S. M. Coxeter. Mathematical Recreations and Essays (1939, page 13); 13th ed. New York: Dover, pp. 14-15, 1987. Gardiner, Anthony, and A. D. Gardiner. Discovering mathematics: The art of investigation. Oxford University Press, 1987. G. H. Hardy, A Mathematician's Apology (Cambridge Univ. Press, 1940, reprinted 2000), pp. 104-105 (describes this problem as having "nothing in [it] which appeals much to a mathematician"). D. Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986. LINKS Ray Chandler, Table of n, a(n) for n = 1..10000 (first 400 terms from Vincenzo Librandi) Martin Beech, A Computer Conjecture of a Non-Serious Theorem, Mathematical Gazette, 74 (No. 467, March 1990), 50-51. Patrick De Geest, Palindromic Products of Integers and their Reversals D. J. Hoey, Palintiples D. J. Hoey, Palintiples [Cached copy] Benjamin V. Holt, Some General Results and Open Questions on Palintiple Numbers, INTEGERS, Electronic J. of Combinatorial Number Theory, Vol. 14, Paper A42, 2014. Benjamin V. Holt, Derived Palintiple Families and Their Palinomials, arXiv:1410.2356 [math.NT], 2014. Benjamin V. Holt, Families of Asymmetric Palintiples Constructed from Symmetric and Shifted-Symmetric Palintiples, arXiv:1412.0231 [math.NT], 2014. L. H. Kendrick, Young Graphs: 1089 et al., arXiv:1410.0106 [math.NT], 2014. L. H. Kendrick, Young Graphs: 1089 et al., J. Int. Seq. 18 (2015) 15.9.7. Leonard F. Klosinski and Dennis C. Smolarski, On the Reversing of Digits, Math. Mag., 42 (1969), 208-210. Lara Pudwell, Digit Reversal Without Apology, Mathematics Magazine, Vol. 80 (2007), pp. 129-132. N. J. A. Sloane, 2178 And All That, Fib. Quart., 52 (2014), 99-120. R. Webster and G. Williams, On the Trail of Reverse Divisors: 1089 and All that Follow, Mathematical Spectrum, Applied Probability Trust, Sheffield, Vol. 45, No. 3, 2012/2013, pp. 96-102. Eric Weisstein's World of Mathematics, Reversal FORMULA If reverse(n) = k*n in base 10, then k = 1, 4 or 9 [Klosinski and Smolarski]. Hence A008919 is the union of A001232 and A008918. - David W. Wilson a(n) = 99*A061852(n). - M. F. Hasler, Oct 17 2022 MAPLE P:=proc(q) local a, b, n; for n from 1 to q do a:=n; b:=0; while a>0 do b:=b*10+(a mod 10); a:=trunc(a/10); od; if type(b/n, integer) then if b/n>1 then print(n); fi; fi; od; end: P(10^10); # Paolo P. Lava, Jul 29 2014 MATHEMATICA Reap[ Do[ If[ Reverse[ IntegerDigits[n]] == IntegerDigits[4*n], Print[n]; Sow[n]]; If[ Reverse[ IntegerDigits[n + 11]] == IntegerDigits[9*(n + 11)], Print[n + 11]; Sow[n + 11]], {n, 78, 2*10^10, 100}]][[2, 1]] (* Jean-François Alcover, Jun 19 2012, after David W. Wilson, assuming n congruent to 78 or 89 mod 100 *) okQ[t_]:=t==Reverse[t]&&First[t]!=0&&Min[Length/@Split[t]]>1; Sort[ Flatten[ {99#, 198#}&/@Flatten[Table[FromDigits/@Select[Tuples[ {0, 1}, n], okQ], {n, 10}]]]] (* Harvey P. Dale, Jul 03 2013 *) PROG (Haskell) a008919 n = a008919_list !! (n-1) a008919_list = [x | x <- [1..], let (x', m) = divMod (a004086 x) x, m == 0, x' > 1] -- Reinhard Zumkeller, Feb 03 2012 (PARI) is_A008919(n, r=A004086(n))={n>r && n%r==0} \\ M. F. Hasler, Jun 23 2019 CROSSREFS Cf. A001232 (9k = R(k)), A004086 (R(n): reverse), A008918 (4k = R(k)), A214927, A103609 (Fibonacci([n/2])). Reversals are in A031877. Sequence in context: A175698 A110819 A071685 * A110843 A354256 A319570 Adjacent sequences: A008916 A008917 A008918 * A008920 A008921 A008922 KEYWORD nonn,base,nice AUTHOR EXTENSIONS Corrected and extended by David W. Wilson Aug 15 1996, Dec 15 1997 STATUS approved

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Last modified November 29 18:13 EST 2022. Contains 358431 sequences. (Running on oeis4.)